In this section, we focus on refining the parameterizations employed for the basic model (Section 3.1.), namely, the parameterizations of the albedo, the outgoing IR flux and the net horizontal energy transport, because they reveal various shortcomings. In the refined EBM we try to overcome some of them. However, we have to keep in mind that it is the degree of the model's spatial resolution in the first instance which sets natural bounds to the various refinements. In order to minimize the use of indices in this section, we will use z = sin 0 as the independent variable to express the dependence on latitude of the equations to be set up.
(a) Albedo
We begin with the ice-albedo parameterization employed by Sellers (1969). He arrived at the value b =
-
0.009"c - ~
by comparing observed zonal albedos and temperatures at similar latitudes in the northern and southern hemisphere. But as pointed out by Gal-Chen and Schneider (1976), the climatology of the two hemispheres differs. In particular, there are differences in cloud amount between the two hemispheres which should introduce a spurious effect into such a zonal albedo comparison. Furthermore, Lian and Cess (1977) argued that the albedo for the ice-covered portion of the hemisphere is greater relative to that of the ice-free portion, not only because of the ice cover, but also as a consequence of the albedo being dependent upon solar zenith angle. In particular, the albedo of clouds is greatly enhanced a t high latitudes by this effect. They showed that the neglect of zenith angle dependence leads to an overestimate of the ice-albedo feedback in the BudykwSellers type of models.For these reasons, we decided to follow Lian and Cess' albedo parameterization which accounts for zenith angle effects as well as the influence of latitudinal variations in cloud amount. They employed the latitudinal variation of zonal albedo to determine the dependence of zonal albedo upon surface temperature:
where a, a, and a, denote the zonal effective albedo, the cloudy-sky albedo and the clear-sky albedo, while A,(z) represents the zonal cloud fraction taken to be constant here (cf. Table 2). corrections of the northern hemisphere (which Lian and Cess considered only) also to the southern hemisphere. In their paper, - Lian and Cess explain in detail how these values are derived. We also assume -
aa,
a T = 0 for latitudes between 40'N and 40"s.
The total derivative - da, is determined from annual zonal data for a,(x) (Vonder Haar dT surface temperature (through a,) and zenith angle. Furthermore, they indicate the hemispherical percentages of the ice-albedo effect (74% for NH, 78% for SH) which are being screened by clouds for the cloud-covered portions of the hemispheres. From Eq.
(49) and Eqs. (51a), (51b), we obtain
Table 2. Mean annual zonal data of the refined EBM. Zone Latitude 1 90-80' N 2 80-70' N 3 70-60'N 4 60-50' N 5 5040'N 6 40-30' N 7 30-20' N 8 20-10' N 9 10- 0'N 10 0-10' S 11 10-20' S 12 20-30' S 13 30-40' S 14 40-50' S 15 50-60' S 16 60-70' S 17 70-80' S 18 80-90' S
A '1 0.55 0.61 0.64 0.64 0.57 0.47 0.41 0.44 0.51 0.50 0.47 0.47 0.54 0.65 0.79 0.77 0.56 0.47
a, 2)
- -.-
dp s)5
4) da,a& -
5) d~ap
d~a~ a~
cr 1 in-
1 1 1 ln'C
In'C
in ' C 0.520 -0.0237 - 0.0007-
0.0230 - 0.0136 0.3590 0.411-
0.0203-
0.0010-
0.0192-
0.0105 0.4145 0.304 -0.0128-
0.0018-
0.0110-
0.0058 0.4225 0.226 -0.0071-
0.0027 - 0.0044-
0.0023 0.4121 0.201 -0.0036-
0.0027 -0.0008 - 0.0005 0.3613 0.176 0.0000 0.0000 0.0000 0.0000 0.3090 0.170 0.0000 0.0000 0.0000 0.0000 0.2720 0.142 0.0000 0.0000 0.0000 0.0000 0.2480 0.150 0.0000 0.0000 0.0000 0.0000 0.2540 0.588 0.0000 0.0000 0.0000 0.0000 0.2410 0.528 0.0000 0.0000 0.0000 0.0000 0.2360 0.339 0.0000 0.0000 0.0000 0.0000 0.2510 0.220 0.0000 0.0000 0.0000 0.0000 0.2960 0.163 -0.0057-
0.0027-
0.0030-
0.0015 0.3726 0.140 -0.0098-
0.0027-
0.0071-
0.0027 0.4339 0.136 -0.0110 - 0.0018 - 0.0092 - 0.0037 0.4877 0.140 -0.0060 - 0.0010 - 0.0050 - 0.0028 0.5191 0.143 - 0.0047 - 0.0007-
0.0040-
0.0025 0.5103 IR7) in - W m2 176.3 178.6 188.0 200.3 217.7 239.1 255.8 258.4 252.7 264.2 263.9 258.5 245.2 226.2 204.3 189.4 167.0 152.1 z8) HT9) in m in-
W m2 137 - 95.3 220-
79.9 202 - 58.7 296 - 35.1 382 - 13.7 496 10.7 366 28.2 146 33.5 158 34.6 154 33.6 121 28.5 156 19.6 106 4 .O 5 - 17.7 5-
38.7 388 - 60.5 1420 - 108.0 2272-
129.8Table 2 continued: 1) Cloud cover fraction; from Cesa (1976). 2) Clear-sky albedo; from Cess (1976). 3) Zenith angle correction; from Lian and Cess (1977). 4) Dependence of clear-sky albedo on surface temperature; from Eq. (50). 5) Dependence of effective albedo on surface temperature; from Eqs. (52a), (52b). 6) From Eq. (53) when tuned against observations. 7) Calculated IR flux; from Eqs. (56a), (56b). 8) Mean height of latitude belt i above aea surface level; from Sellers (1969). 9) Calculated net horizontal energy transport (cf. Table 1) with surface temperatures corrected to sea surface level according to Eq. (57) and with 7 = 2.99 W m-2 'c-I from a least-squares fit.
Similar to Section 3.1., the zonal albedo is then determined from
where the temperature-independent parameter a(z) is tuned to fit the observed zonal planetary albedos. Note that the dependency of the effective albedos on T and p is substituted by a dependency on z. As can be seen from Table i!?, except for the two most northern latitude belts, the absolute values of
, aa ,
are significantly lower thanV 1
Sellers' value of
o.oo~'c-'
in his albedo parameterization. This feature leads to an increased global stability (Warren and Schneider 1979, and cf. also Section 4.1.).Following Lian and Cess, we also do not assume the cloud cover to depend on dA
,
temperature. Observationally, -
dT is positive for the seasonal cycle in some latitude zones and negative in others. ~ h u s it is not clear a t all how cloud cover would change with a change in temperature (Warren and Schneider 1979).
Finally, for use in our refined EBM, Eq. (53) is rewritten in a time-dependent fashion with respect to a change in surface temperature
where - aa is taken to be constant in time.
a T
(6) Infrared jiuz
Cess (1976) assumed the outgoing IR flux a t the top of the atmosphere also to be a linear function of surface temperature. He re-evaluated the linear parameterization using more recent IR data from satellites and cloudiness data. By employing annual and zonal average data for the northern and southern hemisphere separately, he found
(in W rn-2) from a least-squares fit where A, is the fractional cloud cover mentioned
above. Following Warren and Schneider (1979), the fourth term on the right-hand side can be neglected because its contribution to the entire right side does not exceed 1.5%. Thus employing IR(z) = A
+
BT(z)+
kA,(z) to each hemisphere where k is an(cf. Table and Figure 5). As in Section 3.1., we also assume Eqs. (56a), (56b) to be applicable in the form AIR(z,t) = AA(t)
+
BAT(z,t) when used for a climatic change simulation.(c) Horizontal energy transport
Budyko's (1969) formula of the net horizontal energy transport has basically been retained. We have only introduced Sellers' (1969) height corrections (z(z); cf. Table 2) and lapse rate (0.0065'Cm-~) to relate our surface temperatures (ST) to sea surface level (SST) according to
for which we found 7 = 2.99
w
m-2 'c-I from a least-squares fit and 15.86'C for today's global average temperature on sea surface level (cf. Figure 6). Being aware of the fact that Budyko's parameterization of the net horizontal energy transport will be the first one to be changed completely as soon as our EBM approaches a finer subdivision (atmosphere, land, ocean), we found Budyko's refined parameterization satisfactory enough with respect to our present level of modeling. A least-squares fit of a thermal-diffusive transport with a constant diffusion coefficient (Eq. (33)) proved even worse for both temperature levels.4. RESULTS
In this chapter we present the results of simulations from four different energy balance models under various radiative forcing conditions. The energy balance models are the two-hemispherical EBM of IMAGE (Section 3.2.), the two-hemispherical and the 18-latitudinal version of the basic EBM (Section 3.3.2.), and the 18-latitudinal version of the refined EBM (Section 3.4.). The EBM in the climate module of IMAGE is coupled to an advective-diffusive deep ocean and calculates the hemispherical changes in surface temperature. This coupling with the deep ocean is the main difference between the EBM of IMAGE and the other energy balance models mentioned. It has a far-reaching consequence in that the radiative forcing is partly being taken up by the deep ocean which by its long turnover time damps the response of the climate system to radiative forcing whereas the energy balance models set up in this paper respond on a much shorter time scale.
Both transient and time-dependent simulations have been performed with all four energy balance models. As already explained (Section 3.3.), the first refers to a simulation with an instantaneous doubling of atmospheric C02 concentration a t the beginning; the model is then run for some time period to approach its equilibrium.
The second, the time-dependent simulation experiment, represents a model run under gradually increasing greenhouse gas concentrations, implying an increasing radiative forcing scenario (scenario A of the IPCC (1990)). In order to compare all four energy balance models we use the globally averaged change in surface temperature as a reference.